We note that the coordinates of the point B must satisfy the equation of the line, since it lies on the line:

\[{k_2} = mh + b\]

We further note that the y coordinate of A must be greater than the y coordinate of B, since A lies aboveB. Thus, we can say that

\[{k_1} > {k_2}\]

This further means that

\[{k_1}> mh + b\]

We therefore note that every point which lies in the upper half-plane will satisfy the linear inequation

\[y > mx + b\]

Similarly, every point lying in the lower half-plane will satisfy the linear inequation

\[y < mx + b\]

And of course, any point lying exactly on the line will satisfy the linear equation

\[y = mx + b\]

We note the following two special cases:

(1) Horizontal Line

If a line is horizontal, we have seen that it will have an equation of the form \(y= k,\) where k is a constant. In this case, the upper half-plane will correspond to the inequation \(y > k,\) while the lower half-plane will correspond to the inequation \(y < k\). This is shown in the following figure:

(2) Vertical Line

If a line is vertical, we have seen that it will have an equation of the form \(x= k\) where k is a constant. In this case, the right half-plane will correspond the inequation \(x > k\), while the left half-plane will correspond to the inequation \(x < k\). This is shown in the following figure:

Example 1: Plot the graph of the line \(x - 2y = 1\), and find out the inequations corresponding to the upper and lower half-planes.

Solution: First, we find out any two points on the line and draw the graph: